Partitioned kronecker products of matrices and applications

Some generalized commutation matrices are defined and used to establish relationships between π-products and Kronecker products. These are applied to obtain expectations of π-products of random vectors and matrices.     

A note on the information content of a consistent pairwise comparison judgment matrix of an AHP decision maker

A decision maker using the Analytic Hierarchy Process (AHP) could be consistent, and still provide no information in the resulting vector of priorities. An extreme example would be a pairwise comparison judgment matrix filled with 1s which is totally consistent under the various definitions of consistency, but has provided no information about the prioritization of alternatives resulting from the decision maker's judgments. In this paper, the quality of a consistent decision maker's judgments using the Analytic Hierarchy Process is placed in the context of the entropy of the resulting vector of priorities. Indeed, it is the purpose of this paper to provide a formal definition of this notion ofentropy of a priority vector, and to provide a framework for a quantitative measurement of the information content of consistent pairwise comparison judgment matrices of a decision maker who is using the Analytic Hierarchy Process. We will prove that the entropy of the vector of priorities for consistent matrices follows a normal distribution and discuss some general considerations of this result.     

Robust M-estimators of multivariate location and scatter in the presence of asymmetry

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived.     

Linear generalizations of various matric-t distributions

The concept of a matric-t variate is extended to cases where the positive (definite) part of the variate, which is usually Wishart distributed independently of the normal part, is a linear sum of positive (definite) variates with positive coefficients. These distributions and their quadratic forms are of importance i.a, for the exact solution to the multi¬variate Behrens-Fisher problem. A few useful identities con¬cerning the invariant polynomials with matrix arguments are derived     

A structured methodology to implement judiciously the right JIT tactics

In order to enhance the competitiveness and efficiency of manufacturing operations, many companies arc looking at implementing key strategic technologies. Two of the most predominant programs are ‘just-in-time’ and ‘synchronous organizations’. However, their impacts on improving quality and reducing time-to-market have been mixed. It is not very clear why, in some cases, results are poor, when in a similar situation, programmes have proved to work well. There are conflicting reasons reported for such a discrepancy. Some have argued that during J IT implementation either a right mix of tactics was not selected at the outset or the process was not carefully monitored to see whether a mid-course correction or change in tactics was necessary. In order to (a) protect the manufacturing and strategic teams from making the same/similar mistakes and (b) sustain a series of successful activity throughout during the strategic implementation cycle, this paper outlines a structured methodology. The method utilizes a matrix-based procedure to dynamically (over time) measure the effectiveness of a line of JIT tactics against the organization's principles and objectives. The operating procedure suggests first (a) using a method for monitoring the changing conditions of market and business and then (b) using the metrics to guide the management with a new line of tactics that might have better impact on the newly aligned company goals. In an effort to help managers and engineers decide on a proper line of tactics to implement JIT, a line of JIT quality matrices (JQM) is developed. JQM provides a framework to guide group managers to ‘plan, pick and choose’ a set of effective JIT techniques. An approach similar to quality function deployment (QFD) is used to generate the JIT house and their corresponding JQM matrices. With the JQM-based structured methodology, managers can design the best line of JIT strategy blended with JIT theory and adapted to the manufacturing environments in which it is expected to operate.     

UNCONSTRAINED PILOT SELECTORS FOR SMOOTHED CROSS‐VALIDATION

Two of the most useful multivariate bandwidth selection techniques are the plug‐in and cross‐validation methods. The smoothed version of the cross‐validation method is known to reduce the variability of its non‐smoothed counterpart; however, it shares with the plug‐in choice the need for a pilot bandwidth matrix. Owing to the mathematical difficulties encountered in the optimal pilot choice, it is common to restrict this pilot matrix to be a scalar multiple of the identity matrix, at the expense of losing the flexibility afforded by the unconstrained approach. Here we show how to overcome these difficulties and propose a smoothed cross‐validation selector using an unconstrained pilot matrix. Our numerical results indicate that the unconstrained selector outperforms the constrained one in practice, and is a viable competitor to unconstrained plug‐in selectors.     

ON ESTIMATION OF THE POPULATION SPECTRAL DISTRIBUTION FROM A HIGH‐DIMENSIONAL SAMPLE COVARIANCE MATRIX

Sample covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the sample size, the sample covariance matrix demonstrates significant deviations from the underlying population covariance matrix. There is an urgent need to develop new estimation tools in such cases with high‐dimensional data to recover the characteristics of the population covariance matrix from the observed sample covariance matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross‐validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.     

Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application

Let X _n = (x _{i j} ) be a k ×n data matrix with complex‐valued, independent and standardized entries satisfying a Lindeberg‐type moment condition. We consider simultaneously R sample covariance matrices , where the Q _r 's are non‐random real matrices with common dimensions p ×k (k ≥p ). Assuming that both the dimension p and the sample size n grow to infinity, the limiting distributions of the eigenvalues of the matrices { B _{n r} } are identified, and as the main result of the paper, we establish a joint central limit theorem (CLT) for linear spectral statistics of the R matrices { B _{n r} }. Next, this new CLT is applied to the problem of testing a high‐dimensional white noise in time series modelling. In experiments, the derived test has a controlled size and is significantly faster than the classical permutation test, although it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix (R = 1).